A current through a wire varies with time, as \( I = \alpha t + \beta t^{2} \) where \( \alpha = 10\,\mathrm{A\,s^{-1}} \) and \( \beta = 4\,\mathrm{A\,s^{-2}} \). How much charge will cross through a section of the wire in 12 s?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 3024 C
Electric current is the rate of flow of charge, \( I = dq/dt \), so the charge that crosses a section is the integral of the current over time. The current is not constant here, so it cannot be multiplied by 12 s.
\( \displaystyle q = \int_{0}^{12} (\alpha t + \beta t^{2})\,dt = \frac{\alpha t^{2}}{2} + \frac{\beta t^{3}}{3}\Big|_{t=12} \)
\( \displaystyle q = \frac{10\times144}{2} + \frac{4\times1728}{3} = 720 + 2304 = 3024\,\mathrm{C} \)